Abstract
For a sequence of i.i.d. d-dimensional random vectors with independent continuously distributed coordinates, say that the nth observation in the sequence sets a record if it is not dominated in every coordinate by an earlier observation; for , say that the jth observation is a current record at time n if it has not been dominated in every coordinate by any of the first n observations; and say that the nth observation breaks k records if it sets a record and there are k observations that are current records at time but not at time n.
For general dimension d, we identify, with proof, the asymptotic conditional distribution of the number of records broken by an observation given that the observation sets a record.
Fix d, and let be a random variable with this distribution. We show that the (right) tail of satisfies
and
When , the description of in terms of a Poisson process agrees with the main result from Fill (2021) that has the same distribution as , where . Note that the lower bound on implies that the distribution of is not (shifted) Geometric for any .
We show that as ; in particular, in probability as .
Funding Statement
Research supported by the Acheson J. Duncan Fund for the Advancement of Research in Statistics.
Acknowledgments
We thank Ao Sun for providing a simplified proof of Lemma 3.2(i), and Daniel Q. Naiman and Ao Sun for valuable assistance in producing the three figures. We thank Svante Janson for helpful discussions about the details of this paper. We are also grateful for discussions with Persi Diaconis, Hsien-Kuei Hwang, Daniel Q. Naiman, Robin Pemantle, Ao Sun, and Nicholas Wormald. Last but not least, we thank two anonymous reviewers for many helpful suggestions.
Citation
James Allen Fill. "Breaking multivariate records." Electron. J. Probab. 28 1 - 27, 2023. https://doi.org/10.1214/23-EJP968
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